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Type I String Theory
In String Theory, Type I String Theory is one of five consistent Supersymmetryic String Theories in ten dimensions. It is the only one whose strings are unoriented (both orientations of a string are equivalent) and which contains not only Closed Strings, but also Open Strings. The classic 1976 work of Ferdinando Gliozzi, Joel Scherk and David Olive paved the way to a systematic understanding of the rules behind string spectra in cases where only Closed Strings are present via Modular Invariance F. Gliozzi, J. Scherk and D.I. Olive, ``Supersymmetry, Supergravity Theories And The Dual Spinor Model,'' Nucl. Phys. B122 (1977) 253. but did not immediately lead to similar progress for models with Open Strings.. Angelantonj and A. Sagnotti, ``Open strings,'' Phys. Rept. 1 ) 339 arXiv:hep-th/0204089. As first proposed by Augusto Sagnotti in 1987, the Type I String Theory can be obtained as an Orientifold of Type IIB String Theory, with 32 half-D9-Branes added in the vacuum to cancel various anamolyies, along with an extra Open String sector. At low energies, Type I String Theory is described by the \mathcal{N} =1 Supergravity (Type I Supergravity) in ten dimensions.. The discovery in 1984 by Michael Green and John H. Schwarz that anomalies in Type I String Theory cancel was a key reason for the First Superstring Revolution.. However, a key property of these models, shown by Augosto Sagnotti in 1992, is that in general the Green-Schwarz mechanism takes a more general form, and involves several two forms in the cancellation mechanism. The relation (Orientifold Projection) between the Type IIB String Theory and the Type I String Theory has a large number of surprising consequences, both in ten and in lower dimensions, that were first displayed by the String Theory group at the University of Rome "Tor Vergata" in the early Nineties. J. Polchinski, S. Chaudhuri and C.V. Johnson, ``Notes on D-Branes,'' arXiv:hep-th/9602052. It opened the way to the construction of entire new classes of string spectra with or without Supersymmetry. Joseph Polchinski's work on D-Branes provided a geometrical interpretation for these results in terms of extended objects (D-Brane, Orientifold). In the 1990s it was first argued by Edward Witten that Type I String Theory with the string coupling constant g_s is equivalent to the Type HO String Theory with the coupling \frac1{g_s} . E. Witten, ``String theory dynamics in various dimensions,'' Nucl. Phys. B443 (1995) 85 arXiv:hep-th/9503124. This equivalence is known as S-Duality. . . . . Action Principle The Action for the Type I String Theory is the RNS Action , . . The Lagrangian Density is given by: \mathcal{L_{RNS}} = {\frac{T }{2}}h^{ab} \left(\partial_a X^\mu \partial_b X^\nu - i\hbar c_0\bar{\psi_\mu }\not\partial\psi^\mu \right)g_{\mu \nu} The Action is then given by: S_{RNS} = \iint\left( {\frac{T }{2}}h^{ab} \left(\partial_a X^\mu \partial_b X^\nu - i\hbar c_0\bar{\psi_\mu }\not\partial\psi^\mu \right)g_{\mu \nu} \sqrt{-\det h_{\alpha\beta}}\right)\mbox{d}^2\xi Projection of states There are two projections which closed string states go through in the Type I string theory: : 1. The GSO Projection is the same as in Type IIB String Theory. For the Neveu-Schwarz Neveu-Schwarz sector, it is, : : \mathsf{\mathcal{P}}_{\operatorname{NS}}^{-}\left| \psi \right\rangle =\left( 1- \right)\left| \psi \right\rangle For the Ramond Ramond sector, it can either be: : \begin{matrix} \mathsf{\mathcal{P}}_{\operatorname{R}}^{-}\left| \right\rangle =\left( 1-\left( \prod\limits_{k=1}^{10} } \right) \right)\left| \right\rangle \\ \mathsf{\mathcal{P}}_{\operatorname{R}}^{-}\left| \right\rangle =\left( 1-\left( \prod\limits_{k=1}^{10} } \right) \right)\left| \right\rangle \\ \end{matrix} : Or it could be: the following: projection: : : \begin{matrix} \mathsf{\mathcal{P}}_{\operatorname{R}}^{+}\left| \right\rangle =\left( 1+\left( \prod\limits_{k=1}^{10} } \right) \right)\left| \right\rangle \\ \mathsf{\mathcal{P}}_{\operatorname{R}}^{+}\left| \right\rangle =\left( 1+\left( \prod\limits_{k=1}^{10} } \right) \right)\left| \right\rangle \\ \end{matrix} For the Ramond Neveu-Schwarz and the Neveu-Schwarz sectors, one may apply the Neveu-Schwarz sector GSO Projection on the Neveu-Schwarz component of the state and any of the Ramond sector GSO Projection on the Ramond component of the State. : 2. The Orientifold Projection needs to be applied on to the RNS states . This is just another way of saying that only those states which are invariant under Worldsheet parity are taken. This is because, since the string should be unoriented, that just means that even if you flip the string in the other direction, it should look pretty much the same. And since the Worldsheet is just a smear of the String through time, and since the string should alwaysk be unoriented, the Worldsheet should also be "unoriented" in the spatial direction. In other words, the theory should be invariant under Worldsheet parity. For the Open strings, the same GSO Projection applies, but there is no Orientifold projection as there is no Type IIB String Theory with open strings at all!. Instead, one simply takes the GSO Projection of the RNS Open string states and then, plonks them in. Orientifold projection of states The states are orientifold projected from the Type IIB String Theory to the Type I String Theory as follows \begin{align} & }=\frac{\left[ }_{+}}, }}_{+}} \right]}{2} \left( 1 \right)\text{ }\left( \operatorname{R}-R \right) \\ & }=\frac{\left[ \hat{d}_{-1/2}^{\mu },\hat{\tilde{d}}_{-1/2}^{\nu } \right]}{2} \left( 1 \right)\text{ }\left( NS-NS \right) \\ & }=\frac }_{+}}\hat{d}_{-1/2}^{\mu }- }}_{+}}\hat{\tilde{d}}_{-1/2}^{\mu }}{2} \left( 1 \right)\text{ }\left( \operatorname{R}-\operatorname{NS},\operatorname{NS}-R \right)\\ \end{align} The same factors are used to project the mass spectrum, as it is a Hagedorn Mass Spectrum Notes *E. Witten, ``String theory dynamics in various dimensions,'' Nucl. Phys. B443 (1995) 85 arXiv:hep-th/9503124. *J. Polchinski, S. Chaudhuri and C.V. Johnson, ``Notes on D-Branes,'' arXiv:hep-th/9602052. *C. Angelantonj and A. Sagnotti, ``Open strings,'' Phys. Rept. 1 ) 339 arXiv:hep-th/0204089. Category:String Theory